无穷维随机分数阶系统全局动力学行为及其算法研究

The study for the global dynamical behavior of infinite-dimensional dynamical system is the one of the most active subject. Because of the development of modern sciences and technologies, the study for the long-term behavior of stochastic fractional dynamical system, especially, the study for the properties of global attractor and global random attractor, has been a key problem in the field of infinite-dimensional nonlinear dynamics. . In this research project, we will lay stress on some global dynamical behavior and numerical arithmetic for the infinite dimensional stochastic fractional dynamical system. This includes the following contents: 1) the study of the existence of global attractors for certain infinite dimensional fractional systems; 2) researching the existence, estimates on the dimension and stochastic stability of the random attractors for some random fractional systems; 3) to explore the numerical approximation algorithm of global attractors and random attractor, namely, to construct the unconditionally stable numerical schemes for systems and then analysis the existence and the convergence of global attractor and global random attractor; 4) The space-time dynamics of neural electrical activity and the responses to external excitations and environmental noise are analysed by choosing several typical nerve impulse transmission models, and then the dynamics and physiological mechanisms of neural information dissemination are explored.. It is the main aim of this project to advance the development of high-dimensional and infinite-dimensional stochastic dynamics and to provide the important dynamical theoretic basis for the study of neuroscience.

无穷维随机动力系统全局动力学行为的研究是当前最活跃的学科前沿之一。但以往的研究多局限于整数阶系统。由于现代科学技术发展的需要，无穷维随机分数阶系统的全局动力学已成为当前非线性动力学研究中一个至关重要的课题。. 本项目研究某些无穷维随机分数阶系统的全局动力学行为、算法及其应用。内容包括：1）无穷维随机分数阶系统全局吸引子和随机吸引子的存在性、维数估计、正则性、随机稳定性等；2）全局吸引子和全局随机吸引子的数值逼理论，即对随机分数阶模型方程建立长时间收敛和无条件稳定的数值格式，分析近似吸引子的存在性和收敛性等；3）利用典型的神经电脉冲传播模型，分析生物系统放电活动的时空动力学模式，从而探索生物系统电信号传播的动力学特性和生理学机理。. 本项目的最终目标是促进无穷维随机分数阶动力系统理论研究的发展，并为生命科学研究的新方法提供重要的动力学理论依据。

本项目执行期间研究了某些无穷分数阶系统的全局动力学行为、算法及其应用。内容包括：1）无穷维分数阶和随机系统全局吸引子和随机吸引子的存在性、维数估计；2）全局吸引子的数值逼理论，即对分数阶模型方程建立长时间收敛和无条件稳定的数值格式，分析近似吸引子的存在性和收敛性等；3）分数阶偏微方程的算法研究。4）利用典型的神经电脉冲传播模型，分析生物系统放电活动的时空动力学模式，从而探索生物系统电信号传播的动力学特性和生理学机理。